Stochastic Control and Differential Games with Path-Dependent Influence of Controls on Dynamics and Running Cost

Abstract

In this paper we consider the functional Itô calculus framework to find a path-dependent version of the Hamilton--Jacobi-Bellman equation for stochastic control problems that feature dynamics and running costs that depend on the path of the control. We also prove a dynamic programming principle for such problems. We apply our results to path-dependence of the delay type. We further study stochastic differential games in this context.

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